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v. t. e. In mathematics, the derivative is a fundamental tool that quantifies the sensitivity of change of a function 's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
Elementary rules of differentiation. Unless otherwise stated, all functions are functions of real numbers (R) that return real values; although more generally, the formulae below apply wherever they are well defined [1][2] — including the case of complex numbers (C). [3]
t. e. In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g. More precisely, if is the function such that for every x, then the chain rule is, in Lagrange's notation, or, equivalently, The chain rule may also be expressed in ...
Calculus. In calculus, the general Leibniz rule, [1] named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if and are n -times differentiable functions, then the product is also n -times differentiable and its n -th derivative is given by where is the binomial coefficient ...
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear ...
f′ (x) A function f of x, differentiated once in Lagrange's notation. One of the most common modern notations for differentiation is named after Joseph Louis Lagrange, even though it was actually invented by Euler and just popularized by the former. In Lagrange's notation, a prime mark denotes a derivative.
Formal derivative. In mathematics, the formal derivative is an operation on elements of a polynomial ring or a ring of formal power series that mimics the form of the derivative from calculus. Though they appear similar, the algebraic advantage of a formal derivative is that it does not rely on the notion of a limit, which is in general ...
Calculus. Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after Francesco Faà di Bruno (1855, 1857), although he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French mathematician Louis François Antoine Arbogast had ...